The invention is in the fields of seismic exploration for oil, gas and other underground resources and of improving time and space sampled measurements similar to those used in seismic exploration. One aspect of the invention pertains to improving seismic traces by transforming a seismogram into tau-p space, filtering or otherwise processing it in tau-p space, and then if desired inverse-transforming it back into its original space. Another aspect pertains to similarly processing other measurements or data.
The ability to process seismograms in tau-p space is desirable because many operations can be carried out in tau-p space to make the seimograms more useful which could not be done practically, or at least conveniently, in the space of the original seismogram, e.g., in time-offset space. Exactness of the transformation into and out of tau-p space is desirable because otherwise artifacts can be introduced by the inexactness of the transformation and be indistinguishable in the inverted seismogram from actual features of the underground formations. The term "exact" and "inexact" as used herein in connection with the forward and inverse transformations refer to accuracy of results which is inherent in the transformation and is independent of the accuracy of any arithmetic techniques or any equipment used to help carry out the transformations. Thus with respect to a particular transformation "exact" means exact within the accuracy of the arithmetic techniques and the equipment used to carry out that transformation.
In the known prior art it is theoretically possible to use a continuous form of the transformation into and out of tau-p space in order to achieved exact inversion, but this is not believed to be practical for the typical seismic survey, where vast amounts of measurements are taken, typically in digital form, and digital computer equipment is used to process them. It is also theoretically possible to obtain exact inversion by directly discretizing the continuous form of the transformation and treating the inversion as a solution to a linear system of equations. This is not believed to be practical because the nature of the transformation would make the processing time prohibitively long for the typical seismic survey. As a result, it is believed that the prior art transformations into and out of tau-p space use discretized forms of the continuous transformation with compromises which can make the processing time manageable but as a consequence can introduce significant artifacts indistinguishable from features of the underground formations. Thus, exact transformations using the known prior art do not appear to be practical, and the practical prior art transformations can be excessively inexact. This has long been known, and a longstanding need is believed to have existed for a way to transform seismograms into and out of tau-p space with high accuracy and yet within a practical span of time.
The invention meets this need by providing a new kind of a discrete transformation into and out of tau-p space which is both exact and practical in terms of processing time. For example, it is believed that the ratio in processing time in the case of transforming a real-life seismogram into and out of tau-p space can be of the order of 1:1000 as between using this invention and using a known prior art technique of exact transformation on the basis of direct discretization of the continuous transform and solving a linear system of equations for the inversion, i.e., that using the same equipment, the invention would take one minute to do what such prior art would take 1000 minutes (16.66 hours) to do. In addition, the invention makes it possible and practical to transform into tau-p space along selected curves rather than straight lines only, and to weight selected lines or slopes so as to enhance or suppress their significance. It is believed that in the prior art such transforms (using curves and or weighting) were not practical because the known inversions introduced unacceptably excessive errors.
Seismic profiling is one of the major techniques for exploring subsurface formations, for example for the purpose of finding and/or evaluating underground resources such as hydrocarbon deposits. Seismograms are often represented as traces in time-offset space, and often correspond to a planar vertical section through the subsurface formations. Typically, but not necessarily, a seismic trace corresponds to a selected source position on the surface, often called a shotpoint, and a selected receiver position relative to the source position, often called an offset. One form of a seismogram is a collection of such traces for a fixed shotpoint placed along the offset axis in accordance with the respective offsets between the respective receiver positions and the shotpoint. The two-way travel times of reflections from subsurface interfaces of layers which have different acoustic properties show on the trace along the time axis. For some purposes, the time axis is converted to a depth axis. Two major types of seismic exploration are horizontal seismic profiling (often called reflection seismics or surface seismics, and including large-offset surface seismics) and vertical seismic profiling (often abbreviated as VSP).
Subsurface reflection seismology involves mapping of subsurface formations on the basis of the arrival times of events reflected from subsurface layers. Seismic energy generated at the surface penetrates the earth's layered media and some of it is reflected at the interfaces between layers. The reflected energy arriving at the surface is measured by receivers (also called detectors or geophones). On land, seismic energy typically is generated by chemical explosions, vibrating machines or thumping devices. Preplaced receivers arranged in an array or "spread" detect the seismic energy which comes back. At sea, or other bodies of water, a source such as an array of air guns towed behind a ship is actuated every few seconds as the ship moves over a predetermined course. The returning seismic energy is picked up by detectors which typically are embedded in a cable (a streamer) trailing the ship.
Vertical seismic profiling, or VSP, is a technique in which a seismic signal generated at or near the surface of the earth is recorded by receivers secured at various depths to the wall of a borehole. While in reflection seismics the receivers are strung along the earth's surface (or along a nearly horizontal cable trailing a ship), in VSP the receiver locations are strung along the borehole axis. Moreover, in VSP the receivers respond both to downgoing and upgoing seismic events, in contrast to reflection seismics, where the receivers typically do not respond directly to downgoing events. The spacing between receiver locations in VSP typically is a small fraction of that used in reflection seismics.
A VSP seismogram is often represented in time-depth space, rather than the time-offset space common in reflection seismics. The depth axis is depth into the earth, and the time axis is time taken by the seismic wave to travel from the surface to a given depth.
The complexity of subsurface structures and the noise and other imperfections of the observed signals make it necessary to process these signals through a number of steps in order to make them more useful for their ultimate purpose of finding and evaluating subsurface resources. In the case of horizontal profiling, those steps typically include both static and dynamic corrections of the receiver outputs. The static corrections can include a source correction and a receiver (detector) correction, the combined effect of which is to put the seismic energy source and the receivers in the same fictitious horizontal plane. In practice, this is done by appropriately time-shifting the respective receiver outputs. The dynamic corrections convert each receiver output to the output which it would have produced if the source and receiver were at the same lateral point, e.g., the point midway between the actual source and receiver positions along the fictitious horizontal plane. In this conversion the traces are considered to be made up of primary reflections, i.e., the receipt of acoustic energy which has traveled from the source directly down to a reflector and then directly up to the receiver. If the layers are horizontal then all the reflection points (depth points) are directly beneath the midpoint between the actual source and receiver positions. If the layers are dipping then the depth points are offset from this midpoint. The dynamic corrections thus depend both on the dip of the reflecting beds and on the velocity function (the respective velocities of the acoustic signal in each layer traversed by the relevant ray paths). Seismic waves have a velocity which is very much dependent on the nature of the propagating medium, and this velocity can change significantly as the waves travel through different layers. If a suitable borehole is available at the right place and can be logged, the velocity function can be estimated from logs (e.g., acoustic logs) or from vertical seismic profiling. In most cases, however, the velocity function must be estimated from surface measurement, using sophisticated processing. Once the velocity function is available, both components of the dynamic correction can be carried out--the "normal moveout correction" which accounts for the separation between source and receiver and the "dip correction" which accounts for the dip of the relevant subsurface layers. The same velocity function can be used to convert the time-offset traces into depth-offset form, to make the seismogram resemble an anatomical section through the subsurface formations.
Each observed signal (recorded trace) can be considered as made up of reflected waves together with various interfering waves and noise. The desired reflected events typically are the primary reflections. In an effort to suppress events other that primary reflections, typically multifold coverage is recorded by laying out a source and a spread of receivers, activating the source and recording the receiver outputs, then moving the entire configuration along the receiver line and repeating the process. If the increment of movement between shots is small as compared to the receiver spacing, considerable redundancy is introduced which allows the subsequent enhancement of the primary events and suppression of multiples (events reflected more than once before reaching a receiver) and some other noise. If all the traces in a prospect are sorted (gathered) into "gathers" such that all traces in a gather have a common midpoint between the source and receiver, the appropriate moveout correction should convert each trace in a given gather into about the same equivalent trace, namely that primary reflection trace which would have been received if the source and receiver were directly at the common midpoint in question. Stated differently, the primary reflections of all the traces after appropriate moveout should tend to be in phase while the multiples and perhaps some of the other noise should be out of phase. If this approach is used for the normal moveout correction first, then it can also be applied to making the appropriate source and receiver and dip corrections by different gathering of the traces. In the alternative, a simultaneous correction can be used. Thus, one output trace for each midpoint can be produced, which is commonly called the stacked trace for that midpoint. This addresses the multiple reflections (and perhaps some other noise), but an additional process, called deconvolution, is typically used for the so-called reverberation components of the traces. After deconvolution, the signals typically are subjected to a process called migration of depropagation, which can be conceptualized as a process of propagating the waves observed at the surface with the receivers backward in time into the earth to reveal the actual spatial position of the subsurface reflection points at depth. This process is also described as the transformation of signals observed at the surface to signals which would have been observed at depth. Another way of conceptualizing this is to think of a process in which it is desirable to go to a given point in the reconstruction of a seismic map and to check if there is a subsurface reflector at that point. To accomplish this, one can go to the seismic data and integrate along a time-distance curve dictated by a background model. If there is a reflector at the given point of interest, the data are affected to the greatest extent.
VSP involves some similar processing of the receiver outputs, as well as some processing unique to the VSP environment, and can give insight into some fundamental properties of the subsurface formations and assist in the structural, stratigraphic and lithological evaluation and interpretation of these formations. For example it can help differentiate between primary reflections and multiples, estimate reflector dip, correlate shear wave reflections with compressional wave reflections, locate fault planes, determine lithological (and hydrocarbon) effects on propagating wavelets, look for reflectors ahead of the drill bit, identify intrabed multiples, measure the velocity function for both compressional and shear waves, and estimate the conversion of compressional to shear and shear to compressional energy modes within the earth.
While seismic records have long been processed and often presented for viewing and evaluation in time-offset or time-depth space, in some cases it is desirable to convert them to tau-p space, where a number of filtering and other processing operations can be carried out more effectively to improve those records. For example, it is often believed desirable to process seismograms in tau-p space in order to isolate ground roll and refractions from reflections, to carry out beam steering by limiting the angle of incidence, to separate P-waves and mode-converted S-waves and to combine multicomponent recordings, as well as to carry out other filtering and signal modifying processes. While in the usual seismogram one axis represents two-way travel time and the other offset along the surface, or one represents travel time and the other depth into the earth, in tau-p space one axis can represent two-way reflection time at zero offset distance and the other can be a set of traces each of which corresponds to a particular "p" or ray parameter. This p parameter can be expressed in different ways, as is known in the art. Each trace is tau-p space can represent a single angle of incidence of the actual seismic signal at the surface. Thus, events can be separated by angle of incidence at the surface, and multiples become periodic in tau-p space. After filtering or other processing in tau-p space, the seismogram can be converted back to the more familiar time-offset or time-depth space.
The transformation to tau-p space can be conceptualized as integration along lines. The integral along a line in time-offset or time-depth space transforms to a point in tau-p space. The succession of points making up a trace in tau-p space can be conceptualized as made up of the respective integrals of a corresponding succession of parallel lines in time-offset or time-depth space. Mathematically, the transformation (if continuous) can be described as the Radon transform, published in 1917 by the German scientist Radon. The transform discussed by Radon is continuous, while in real life a seismogram typically is discrete both temporally and spatially--i.e., the outputs of spaced-apart receivers are typically digitized into time-samples. Thus, seismograms are represented in discrete rather than continuous form. Moreover, the digital computer equipment typically used to process seismic receiver outputs also typically operates on discrete rather than continuous signals. For these reasons, various discrete form approximations of the Radon transform are believed to be used currently to transform time-offset spacde seismograms to tau-p space seismograms, and discrete approximations of the inverse Radon transforms are believed to be used to convert back to time-offset space. To the knowledge of the inventor herein, it was believed prior to this invention that there was no discrete form of the Radon transform which could provide an exactly invertible transformation within a practical time frame in the context of the typical amount of measurements collected in a seismic study. It is believed that the prior art compromises made in order to make the processing time manageable led to artifacts which, given the complexity of the seismic data and of the transformation process itself, are such that no precise answer could be given in important cases as to whether a given feature in the filtered, inverse-transformed seismogram is indeed a feature of the subsurface formation or an artifact due to the inexact transformation. In this context the inexactness of interest is not the normal rounding off of numerical values inherent in digital computer operation, but the fundamental and heretofore believed unavoidable (except by devoting an impractical amount of computer power and time) inexactness of an approximation of a transformation defined only in continuous form. It is believed therefore that a long-felt, but unsolved, need existed to find a way to convert to tau-p space by a discrete transform which is exactly invertible in a practical time frame within the accuracy limits of digital computer equipment. An important aspect of the invention is to meet that need.
The invention makes use of the unexpected discovery that, contrary to previous thought, a discrete form can be defined of a transform to tau-p space which makes the tau-p space seismogram exactly invertible in a practical time span. Both the new transform and the seismogram are in discrete form, and yet what was heretofore thought impossible does in fact occur--the transform is exactly invertible within a practical time span. Using the new technique, a seismogram can be transformed to tau-p space, then transformed back to the space of the original seismogram within a practical time span, and be identical to it within the accuracy limitations of the digital computer equipment employed for the purpose. Of course, the point is to filter or otherwise process a seismogram in tau-p space for the purpose of improving it. As a result, when inverse-transformed the modified seismogram usually would differ from the original seismogram. However, for the first time now one can be assured that the features of the inverse-transformed seismogram are not artifacts due to any inherent inexactness of the transformation and inverse-transformation, even though the transformations can be completed within a practical time span.
In a particular and nonlimiting embodiment, the invention comprises deriving a seismogram of selected subsurface formations, transforming it to tau-p space through a discrete transformation which makes the tau-p space seismogram exactly invertible within a practical time span, and processing the seismogram in tau-p space to enchange desirable features thereof and thereby generate an improved seismogram.
The original seismogram can be derived in any suitable known way, for example by measuring the outputs of suitable receivers and, if desired, processing those outputs through part or all of the appropriate prior art processes to improve it (in part or as much as possible in the prior art). In the alternative, the original seismogram can be recorded in a suitable storage medium and derived by reading it out therefrom. The recorded seismogram also can comprise the outputs of seismic receivers, or those outputs as improved by partial prior art processing (in which case additional prior art processing can be used, if desired, as a part of the deriving step), or it can be a seismogram fully processed by desired prior art techniques such as stacking. The original seismogram can be in time-offset space, in time-depth space, or in some other space transformable to tau-p space.
If desired, the seismogram after transformation can be left in tau-p space. A tangible representation thereof can be provided for evaluation or other utilization, for example by displaying a visible representation thereof on a screen or by printing a hard-copy, or by storing it, for example in a suitable computer memory, for further processing or use.
In the alternative, the seismogram after filtering or other processing in tau-p space can be inverse-transformed from tau-p space using a new, discrete form of an inverse transform in accordance with the invention to thereby generate an improved seismogram in a space such as time-offset space or time-depth space, or some other space into which inversion from tau-p space is possible. Again, a tangible representation of this improved seismogram can be provided, for example by displaying a visible representation thereof on a screen or by printing a hard-copy, or by storing it, for example in a suitable computer memory, for further processing or use.
In a more specific, and still nonlimiting embodiment, the invention comprises deriving a seismogram of selected subsurface formations. The seismogram can be represented as a set of traces each comprising discrete samples of a selected seismic parameter, i.e., as a two-dimensional array of time samples of the seismic parameter. This array, which can be called the original seismogram although in fact it may have been processed in conventional ways, is transformed into tau-p space by converting it to frequency space, for example through a discrete Fourier transform (DFT) process, multiplying the resulting frequency space array by selected transformation matrices, independently for each frequency, to derive respective product arrays which are in frequency space, and inverse transforming the product arrays from frequency space to derive a representation of the seismogram in discrete form in tau-p space, which can be in the form of a two-dimensional array similar in structure to the original seismogram. The resulting seismogram in tau-p can be modified by suitable filtering or other processing in tau-p space. The resulting seismogram is in tau-p space and can be called a modified or filtered seismogram.
Thereafter, and if desired, in order to convert the modified seismogram back to another space, for example that of the original seismogram, the process comprises transforming the modified seismogram into frequency space, for example by using a DFT process, multiplying the frequency space version of the modified seismogram by selected adjoint transformation matrices of the original transformation matrices, frequency-by-frequency, to derive respective adjoint product arrays, then using the adjoint matrices and the original transformation matrices to derive selected matrices for filtering, solving for the array a linear relationship between the array, the matrices for filtering and the adjoint product arrays frequency-by-frequency, and inverse-transforming the last-recited array out of frequency space to thereby generate a seismogram of said selected subsurface formations which corresponds to said modified seismogram in tau-p space but is represented as a set of traces in another space, for example, in the space of the original seismogram.
The terms original seismogram and seismic trace are used herein in a broad sense, and are not limited to the typical time-offset or time-depth representations. For example, these terms can apply to depth-offset representations, and to any other representation which is in a space (or domain) which can be converted to tau-p space in accordance with the techniques described herein, or equivalent techniques. Similarly, the term tau-p space is used both to refer to Radon transform space and to any equivalent space (or domain) resulting from transforming an original discretized representation to a new discretized representation in a new space (or domain) in which each point results from a combination (sum) of values along a corresponding selected curve in the original representation, and the new representation being invertible to the original one. The term "curve" is used in this context to include a narrow strip which can be straight or curved, and the term "point" is used to mean an elemental area or a pixel (picture element). The invention makes it possible to use "curves" which need not be along straight lines, i.e., they can actually bend in a desired manner. In the broadest application of the invention, a "curve" includes any geometrical shape, continuous or disjoint. It also makes it possible to weight selected curves to enhance or suppress the influence of selected parameters of the underground formations on the measurement process.
Although described herein primarily in the context of its application to seismic exploration, the invention is not limited thereto. Thus, in another non-limiting embodiment an original record is made of measurements of a selected parameter of energy (such as electromagnetic or nuclear) propagating in a medium and sampled in time and space (corresponding to measuring the amplitude of seismic energy propagating in a formation, in time and depth or offset). This original record is transformed to a record in tau-p space, as defined above, through a discrete transformation having the characteristics described earlier in respect of application of the invention to seismograms. This tau-p space record is processed to enhance desirable features thereof and, either simultaneously or subsequently, effect an exact discrete inverse transformation from tau-p space to generate an improved record. Finally a tangible representation of the improved record is produced for further processing, inspection or the like.
In another aspect of the invention, an original record is made of data for a selected parameter sampled with respect to two independent variables (such as time and space, or two spatial coordinates) defining an original domain. This original record is then transformed to a different domain in which each point results from a sum of values of the selected parameter taken over a corresponding pattern in the original domain (an example of a pair of domains having this relationship is the time-space and tau-p domain pair). To this end a discrete transformation is used which permits exact discrete inverse-transformation and which in processing in a frequency domain corresponding to one of the independent variables can be completed frequency-to-frequency and in processing in the original domain with respect to the other of the independent variables can utilize a transformation matrix having a block-circulant structure. Processing to enhance desirable features, exact discrete inverse transformation to generate an improved record and production of a tangible representation of the improved record then take place as already described. A practical example of such an application is automatic character recognition to enable mechanized reading of lines of printed or written text.
While processing in frequency space is one way of implementing the invention, it is also possible to carry out another version of the invention without need for frequency space transformation, by using transformation matrices which correspond to those discussed above, but are represented in the spatial domain. Which version is applied in a particular case can depend on factors such as the nature of a particular seismic survey and the available processing equipment.
A key aspect of the invention is the provision of a structure in the forward transformation (into tau-p space) which is chosen to make the inverse transformation exact within practical processing time. An important factor in this is the discovery of a transformation matrix which when in the spatial domain forms a block-circulant matrix and when in the frequency domain makes it possible to process all the needed seismic data one frequency at a time. Because of the block-circulant property, the transformation matrix in the spatial domain need not be stored in full or proceed in full--a single row of blocks can be stored and recirculated to derive all other needed rows. In the frequency domain, the nature of the transformation matrices in practicing the invention is such that one matrix for a given frequency takes care of processing all the needed seismic data relating to that frequency. The transformation matrices in frequency space can be generated as needed, or only a few needed parameters describing them may be stored.
In contrast, in the prior art the known exact discretized transformations into and out of tau-p space can require storage and manipulation of extremely large matrices. Taking as an example a simplistic case of an original seismogram comprising 10 traces of 1000 samples each, a prior art technique can require storing and manipulating a matrix of 10 exp 8 elements. If the prior art technique when inverting from tau-p space is to consider the sought seismogram in time-offset space as a linear system of equations having 10,000 unknowns, it is seen that even in this simplistic case the number of operations to find these unknowns is about 10 exp 8. In contrast, the corresponding number of operations using an example of the invention is about 10 exp 5 similarly extensive operations, with much reduced storage requirements as well. This is the same order of magnitude as in the case of the known prior art in which compromises are made which make the transformation inherently inexact.